On the distribution of the free path length of the linear flow in a honeycomb
| dc.creator | Boca, Florin P. | |
| dc.creator | Gologan, Radu N. | |
| dc.date | 2008-02-07 | |
| dc.date | 2008-07-08 | |
| dc.date.accessioned | 2026-07-07T09:48:31Z | |
| dc.date.available | 2026-07-07T09:48:31Z | |
| dc.description | Let $\ell \geq 2$ be an integer. For each $\eps >0$ remove from $\R^2$ the union of discs of radius $\eps$ centered at the integer lattice points $(m,n$, with $m\nequiv n\mod{\ell}$. Consider a point-like particle moving linearly at unit speed, with velocity $ω$, along a trajectory starting at the origin, and its free path length $τ_{\ell,\eps} (ω)\in [0,\infty]$. We prove the weak convergence of the probability measures associated with the random variables $\eps τ_{\ell,\eps}$ as $\eps \to 0^+$ and explicitly compute the limiting distribution. For $\ell=3$ this leads to an asymptotic formula for the length of the trajectory of a billiard in a regular hexagon, starting at the center, with circular pockets of radius $\eps\to 0^+$ removed from the corners. For $\ell=2$ this corresponds to the trajectory of a billiard in a unit square with circular pockets removed from the corners and trajectory starting at the center of the square. The limiting probability measures on $[0,\infty)$ have a tail at infinity, which contrasts with the case of a square with pockets and trajectory starting from one of the corners, where the limiting probability measure has compact support. | |
| dc.description | 20 pages, 9 figures | |
| dc.identifier | https://arxiv.org/abs/0802.1019 | |
| dc.identifier | http://arxiv.org/abs/0802.1019 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/164245 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.subject | 37A60; 11B57; 11L05; 11P21; 37D50; 82C40; 82D05 | |
| dc.title | On the distribution of the free path length of the linear flow in a honeycomb | |
| dc.type | text |